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# Applied 40S April 13, 2009

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More on working with & understanding z-scores.

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### Applied 40S April 13, 2009

1. 1. All Together Now a statistics workshop Let's put our heads together by ﬂickr user Normski's
2. 2. The 3 meanings of a Shaded Normal Curve area % P(E) zlow zhigh The shaded area under a normal curve between two z-scores is interpreted, simultaneously, as: • an area (the area under the normal curve) • a percentage (the percentage of all values in a data set that lie between two particular z-scores) • a probability (the probability that a particular z-score falls between two given z-scores)
3. 3. Case 1(b): Calculate the Percentage of Passing Scores The mean mark for a large number of students is 69.3 percent with a standard deviation of 7 percent. What percent of the students have a passing mark if they must get 60 percent or better to pass? Assume that the marks are normally distributed. HOMEWORK
4. 4. Case 2(a): Calculate the Percentage of Scores Between Two Z-Scores HOMEWORK (Case 2) If we know two z-scores of a standard normal distribution, we can ﬁnd the percentage of scores that lie between them. The procedure is similar to that used in the previous examples. Sample question(s):What percent of scores lie between z = 0.87 and z = 2.57? OR What is the probability that a score will fall between z = 0.87 and z = 2.57? OR Find the area between z = 0.87 and z = 2.57 in a standard normal distribution.
5. 5. Case 2(b): Calculate the Percentage of Scores Between Two Z-Scores HOMEWORK Find the probability of getting a z-score less than 0.75 in a standard normal distribution.
6. 6. Case 3(b): Find the Z-Value that Corresponds to a Given Probability HOMEWORK What is the z-score if the probability of getting less than this z-score is 0.750?
7. 7. The 3 meanings of a Shaded Normal Curve P(E) zlow zhigh The shaded area under a normal curve between two z-scores is interpreted, simultaneously, as: • an area (the area under the normal curve) • a percentage (the percentage of all values in a data set that lie between two particular z-scores) • a probability (the probability that a particular z-score falls between two given z-scores)
8. 8. Case 3(a): Find the Z-Score that Corresponds to a Given Probability HOMEWORK If we know the probability of an event, we can ﬁnd the z-score that corresponds to this probability. This is the reverse of what we did in Case 2. Sample question: What is the z-score if the probability of getting more than this z-score is 0.350?
9. 9. A club called quot;The Beanstalk Clubquot; has a minimum height requirement of 5'10''. Women in North America have a mean height of 5' 5.5'', and a standard deviation of 2.5''. What percentage of women are eligible? Assume that the heights of women normally distributed. Include a sketch with your answer.
10. 10. An orange producer who calls himself Doctor Juice grows an exclusive variety of oranges which are sorted into three categories and sold at different prices. Description Size Price per orange Small less than 75mm 12 cents Jumbo largest 12% 45 cents Regular all others 35 cents The diameters of the oranges are distributed normally with a mean of 84 mm and a standard deviation of 12 mm. HOMEWORK (a) What percent of the oranges are sorted into the small category? (b) What is the minimum diameter (rounded to the nearest millmeter) of a Jumbo Orange? (c) What is the expected income from 2000 unsorted oranges,
11. 11. Forty students measured the width of the gym, and wrote their measurements in centimetres, rounded to the nearest cm. The measurements are recorded on the table below. HOMEWORK 2251 2249 2250 2247 2253 2248 2249 2253 2254 2247 2250 2253 2248 2255 2249 2249 2250 2251 2252 2250 2249 2250 2247 2250 2250 2252 2253 2255 2254 2248 2248 2242 2249 2245 2251 2246 2250 2246 2251 2246 Draw a histogram of the data. Using the properties of a Normal Distribution, determine if the data is approximately normal.
12. 12. A college aptitude test is scaled so that its scores approximate a normal distribution with a mean of 500 and standard deviation of 100. (a) Find the probability that a student selected at random will score 800 or more points. HOMEWORK (b) Find the score x, such that 76 percent of the students have a score less than x.
13. 13. The weights of babies born in a certain hospital average 8 lb 1 oz, with a standard deviation of 12 oz. Assume that the weights are normally distributed. HOMEWORK (a) Find the percentage of babies with a birth weight between 7 and 9 pounds. (b) Find the weight, W, such that the percentage of babies with a birth weight greater than W is 60 percent. (c) Find the weight, W, such that the percentage of babies with a birth weight less than W is 25 percent.